Barnes’ multiple Bernoulli and generalized Barnes’ multiple Frobenius-Euler mixed-type polynomials

Dae San Kim,Takao Komatsu,Hyuck In Kwon,Taekyun Kim

doi:10.1186/1687-1847-2014-238

Dae San Kim, Takao Komatsu + Show 2 more

Open Access

https://doi.org/10.1186/1687-1847-2014-238

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Abstract

In this paper, by considering Barnes’ multiple Bernoulli polynomials as well as generalized Barnes’ multiple Frobenius-Euler polynomials, we define and investigate the mixed-type polynomials of these polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities. MSC:05A15, 05A40, 11B68, 11B75, 33E20, 65Q05.

Highlights

BHn(x) = BHn(x|a; b; λ; μ) = BHn(x|a, . . . , ar; b, . . . , bs; λ, . . . , λs; μ, . . . , μs) called Barnes’ multiple Bernoulli and generalized Barnes’ multiple Frobenius-Euler mixed-type polynomials, whose generating function is given by r i=
From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities
L|p(x) is the action of the linear functional L on the polynomial p(x), and we recall that the vector space operations on P∗ are defined by L + M|p(x) = L|p(x) + M|p(x), cL|p(x) = c L|p(x), where c is a complex constant in C

Summary

Introduction

Μs) called Barnes’ multiple Bernoulli and generalized Barnes’ multiple Frobenius-Euler mixed-type polynomials, whose generating function is given by r i=. Μr) are called Barnes’ multiple Bernoulli and generalized Barnes’ multiple Frobenius-Euler mixed-type numbers. Μs) are called generalized Barnes-type Euler polynomials. By considering Barnes’ multiple Bernoulli polynomials as well as generalized Barnes’ multiple Frobenius-Euler polynomials, we define and investigate the mixedtype polynomials of these polynomials. L|p(x) is the action of the linear functional L on the polynomial p(x), and we recall that the vector space operations on P∗ are defined by L + M|p(x) = L|p(x) + M|p(x) , cL|p(x) = c L|p(x) , where c is a complex constant in C. Sheffer sequences are characterized in the generating function [ , Theorem .

The Sheffer identity Theorem

A relation with higher-order Frobenius-Euler polynomials Theorem nn